Magnetic Resonance for Arbitrary Spins for Arbitrary B Field Directions

In 1952, Schwinger published On Angular Momentum, which indicates arbitrary angular momentums can be reduced to harmonic oscillators. In 1966, Gottfried solved the Schroedinger equation for magnetic resonance for arbitrary spins by transforming the system to a rotating frame. At last year's APS meeting, we presented a simple way to calculate the magnetic resonance of arbitrary spins by the rotation wave approximation. The situations above always have a B field along the z-direction and an oscillatory B field in the x-y plane. However, we don’t have a theory to calculate the B field's case in arbitrary directions. In here, we provide a way to calculate magnetic resonance for arbitrary directions and arbitrary B fields precisely. First, we refresh the rotating wave approximation method we used at last year's APS March meeting. Then, we will use the effective Hamiltonian we calculated to compare with the Hamiltonian without using the rotating wave approximation to find the (Liu-Klemm) symmetric and skew-symmetric relations for matrices. Then, we will use the relationship we found to solve for the magnetic resonance for arbitrary spins with fields in arbitrary directions.

Interestingly, we also will show the components inside the Rabi frequency. After that, we will combine the result for every single direction back to the combined direction as the previous direction and show the matching result. At last, we will show magnetic resonance for arbitrary spins in each combined direction.